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In mathematics, the harmonic series is the divergent infinite series:

$$\sum_{n=1}^\infty\,\frac{1}{n} \;\;=\;\; 1 \,+\, \frac{1}{2} \,+\, \frac{1}{3} \,+\, \frac{1}{4} \,+\, \frac{1}{5} \,+\, \cdots.\!$$

Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength. Every term of the series after the first is the harmonic mean of the neighboring terms; the phrase harmonic mean likewise derives from music.

History

The fact that the harmonic series diverges was first proven in the 14th century by Nicole Oresme, but this achievement fell into obscurity. Proofs were given in the 17th century by Pietro Mengoli, Johann Bernoulli, and Jacob Bernoulli.

Historically, harmonic sequences have had a certain popularity with architects. This was so particularly in the Baroque period, when architects used them to establish the proportions of floor plans, of elevations, and to establish harmonic relationships between both interior and exterior architectural details of churches and palaces.

The harmonic series is counterintuitive to students first encountering it, because it is a divergent series though the limit of the nth term as n goes to infinity is zero. The divergence of the harmonic series is also the source of some apparent paradoxes. One example of these is the "worm on the rubber band". Suppose that a worm crawls along a 1 metre rubber band and, after each minute, the rubber band is uniformly stretched by an additional 1 metre. If the worm travels 1 centimetre per minute, will the worm ever reach the end of the rubber band? The answer, counterintuitively, is "yes", for after n minutes, the ratio of the distance travelled by the worm to the total length of the rubber band is

$$\frac{1}{100}\sum_{k=1}^n\frac{1}{k}.$$

Because the series gets arbitrarily large as n becomes larger, eventually this ratio must exceed 1, which implies that the worm reaches the end of the rubber band. The value of n at which this occurs must be extremely large, however, approximately e100, a number exceeding 1040. Although the harmonic series does diverge, it does so very slowly.

Another example is: given a collection of identical dominoes, it is clearly possible to stack them at the edge of a table so that they hang over the edge of the table. The counterintuitive result is that one can stack them in such a way as to make the overhang arbitrarily large, provided there are enough dominoes.
Divergence

There are several well-known proofs of the divergence of the harmonic series. Two of them are given below.
Comparison test

One way to prove divergence is to compare the harmonic series with another divergent series:

\begin{align} & 1 \;\;+\;\; \frac{1}{2} \;\;+\;\; \frac{1}{3} \,+\, \frac{1}{4} \;\;+\;\; \frac{1}{5} \,+\, \frac{1}{6} \,+\, \frac{1}{7} \,+\, \frac{1}{8} \;\;+\;\; \frac{1}{9} \,+\, \cdots \\[12pt] >\;\;\; & 1 \;\;+\;\; \frac{1}{2} \;\;+\;\; \frac{1}{4} \,+\, \frac{1}{4} \;\;+\;\; \frac{1}{8} \,+\, \frac{1}{8} \,+\, \frac{1}{8} \,+\, \frac{1}{8} \;\;+\;\; \frac{1}{16} \,+\, \cdots. \end{align}

Each term of the harmonic series is greater than or equal to the corresponding term of the second series, and therefore the sum of the harmonic series must be greater than the sum of the second series. However, the sum of the second series is infinite:

\begin{align} & 1 + \left(\frac{1}{2}\right) + \left(\frac{1}{4}+\frac{1}{4}\right) + \left(\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\right) + \left(\frac{1}{16}+\cdots+\frac{1}{16}\right) + \cdots \\[12pt] =\;\; & 1 \;\;+\;\; \frac{1}{2} \;\;+\;\; \frac{1}{2} \;\;+\;\; \frac{1}{2} \;\;+\;\; \frac{1}{2} \;\;+\;\; \cdots \;\;=\;\; \infty. \end{align}

It follows (by the comparison test) that the sum of the harmonic series must be infinite as well. More precisely, the comparison above proves that

$$\sum_{n=1}^{2^k} \,\frac{1}{n} \;\geq\; 1 + \frac{k}{2}$$

for every positive integer k.

This proof, due to Nicole Oresme, is considered by some a high point of medieval mathematics. It is still a standard proof taught in mathematics classes today. Cauchy's condensation test is a generalization of this argument.
Integral test

It is possible to prove that the harmonic series diverges by comparing its sum with an improper integral. Specifically, consider the arrangement of rectangles shown in the figure to the right. Each rectangle is 1 unit wide and 1 / n units high, so the total area of the rectangles is the sum of the harmonic series:

$$\begin{array}{c} \text{area of}\\ \text{rectangles} \end{array} = 1 \,+\, \frac{1}{2} \,+\, \frac{1}{3} \,+\, \frac{1}{4} \,+\, \frac{1}{5} \,+\, \cdots.$$

However, the total area under the curve y = 1 / x from 1 to infinity is given by an improper integral:

$$\begin{array}{c} \text{area under}\\ \text{curve} \end{array} = \int_1^\infty\frac{1}{x}\,dx \;=\; \infty.$$

Since this area is entirely contained within the rectangles, the total area of the rectangles must be infinite as well. More precisely, this proves that

$$\sum_{n=1}^k \, \frac{1}{n} \;>\; \int_1^{k+1} \frac{1}{x}\,dx \;=\; \ln(k+1).$$

The generalization of this argument is known as the integral test.
Rate of divergence

The harmonic series diverges very slowly. For example, the sum of the first 1043 terms is less than 100. This is because the partial sums of the series have logarithmic growth. In particular,

$$\sum_{n=1}^k\,\frac{1}{n} \;=\; \ln k + \gamma + \varepsilon_k$$

where \gamma is the Euler–Mascheroni constant and \varepsilon_k ~ \frac{1}{2k} which approaches 0 as k goes to infinity. This result is due to Leonhard Euler. He proved also the more striking fact that the sum which includes only the reciprocals of primes also diverges, i.e.

$$\sum_{p\text{ prime }}\frac1p = \frac12 + \frac13 + \frac15 + \frac17 + \frac1{11} + \frac1{13} + \frac1{17} +\cdots = \infty.$$

Partial sums

The nth partial sum of the diverging harmonic series,

$$H_n = \sum_{k = 1}^n \frac{1}{k},\!$$

is called the nth harmonic number.

The difference between the nth harmonic number and the natural logarithm of n converges to the Euler–Mascheroni constant.

The difference between distinct harmonic numbers is never an integer.

No harmonic numbers are integers, except for n = 1.
Related series
Alternating harmonic series
The first fourteen partial sums of the alternating harmonic series (black line segments) shown converging to the natural logarithm of 2 (red line).

The series

$$\sum_{n = 1}^\infty \frac{(-1)^{n + 1}}{n} \;=\; 1 \,-\, \frac{1}{2} \,+\, \frac{1}{3} \,-\, \frac{1}{4} \,+\, \frac{1}{5} \,-\, \cdots$$

is known as the alternating harmonic series. This series converges by the alternating series test. In particular, the sum is equal to the natural logarithm of 2:

$$1 \,-\, \frac{1}{2} \,+\, \frac{1}{3} \,-\, \frac{1}{4} \,+\, \frac{1}{5} \,-\, \cdots \;=\; \ln 2.$$

This formula is a special case of the Mercator series, the Taylor series for the natural logarithm.

A related series can be derived from the Taylor series for the arctangent:

$$\sum_{n = 0}^\infty \frac{(-1)^{n}}{2n+1} \;\;=\;\; 1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \cdots \;\;=\;\; \frac{\pi}{4}.$$

This is known as the Leibniz formula for pi.
General harmonic series

The general harmonic series is of the form

$$\sum_{n=0}^{\infty}\frac{1}{an+b}.\!$$

where a \ne 0 and b are real numbers.

By the comparison test, all general harmonic series diverge. 
P-series

A generalization of the harmonic series is the p-series (or hyperharmonic series), defined as:

$$\sum_{n=1}^{\infty}\frac{1}{n^p},\!$$

for any positive real number p. When p = 1, the p-series is the harmonic series, which diverges. Either the integral test or the Cauchy condensation test shows that the p-series converges for all p > 1 (in which case it is called the over-harmonic series) and diverges for all p ≤ 1. If p > 1 then the sum of the p-series is ζ(p), i.e., the Riemann zeta function evaluated at p.
φ-series

For any convex, real-valued function φ such that

$$\limsup_{u\to 0^{+}}\frac{\varphi(\frac{u}{2})}{\varphi(u)}< \frac{1}{2}$$

the series ∑n≥1 φ(n−1) is convergent.
Random harmonic series

The random harmonic series

$$\sum_{n=1}^{\infty}\frac{s_{n}}{n},\!$$

where the sn are independent, identically distributed random variables taking the values +1 and −1 with equal probability 1/2, is a well-known example in probability theory for a series of random variables that converges with probability 1. The fact of this convergence is an easy consequence of either the Kolmogorov three-series theorem or of the closely related Kolmogorov maximal inequality. Byron Schmuland of the University of Alberta further examined the properties of the random harmonic series, and showed that the convergent is a random variable with some interesting properties. In particular, the probability density function of this random variable evaluated at +2 or at −2 takes on the value 0.124999999999999999999999999999999999999999764…, differing from 1/8 by less than $$10^42$$. Schmuland's paper explains why this probability is so close to, but not exactly, 1/8. The exact value of this probability is given by the infinite cosine product integral $$C_2$$  divided by π.
Depleted harmonic series
Main article: Kempner series

The depleted harmonic series where all of the terms in which the digit 9 appears anywhere in the denominator are removed can be shown to converge and its value is less than 80. In fact when terms containing any particular string of digits are removed the series converges.

Complex logarithm
Lagarias's theorem
Harmonic progression

References

^ George L. Hersey, Architecture and Geometry in the Age of the Baroque, p 11-12 and p37-51.
^ a b Graham, Ronald; Knuth, Donald E.; Patashnik, Oren (1989), Concrete Mathematics (2nd ed.), Addison-Wesley, pp. 258–264, ISBN 978-0-201-55802-9
^ Sharp, R.T. (1954), "Problem 52: Overhanging dominoes", Pi Mu Epsilon Journal: 411–412
^ Sloane's A082912 : Sum of a(n) terms of harmonic series is > 10^n. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ http://mathworld.wolfram.com/HarmonicNumber.html
^ Art of Problem Solving: "General Harmonic Series"
^ "Random Harmonic Series", American Mathematical Monthly 110, 407-416, May 2003
^ Schmuland's preprint of Random Harmonic Series
^ Weisstein, Eric W. “Infinite Cosine Product Integral.” From MathWorld – a Wolfram Web Resource. http://mathworld.wolfram.com/InfiniteCosineProductIntegral.html accessed 11/14/2010
^ Nick's Mathematical Puzzles: Solution 72

Weisstein, Eric W., "Harmonic Series" from MathWorld.
"Harmonic Series" at Springer Encyclopaedia of Mathematics